How Correlation Changes Portfolio Risk

Why Correlation Matters When Holdings Stay Fixed

Two assets can have the same individual volatilities and portfolio weights but produce different portfolio risk because their returns may move together to different degrees. If one asset tends to have a high return when the other does, their joint variation tends to add to portfolio risk. If their returns tend to move in opposite directions, their joint variation can reduce it. The weights and standalone volatilities alone do not determine the portfolio variance.

This tutorial isolates that effect: the asset weights and volatilities are held fixed while correlation changes. The calculations use returns measured over the same period and in the same units. Examples use annual returns and annual standard deviations, with decimal returns in the formulas. Unless stated otherwise, the assets are risky assets in a fully invested portfolio, so the weights sum to one.

Definition: Let \(R_1\) and \(R_2\) be the returns on two assets over the same period, with standard deviations \(\sigma_1\) and \(\sigma_2\). Their correlation is \(\rho_{12}=\operatorname{Cov}(R_1,R_2)/(\sigma_1\sigma_2)\), when both standard deviations are positive. For portfolio weights \(w_1,w_2\), the portfolio return is \(R_p=w_1R_1+w_2R_2\), and its variance is \(w_1^2\sigma_1^2+w_2^2\sigma_2^2+2w_1w_2\rho_{12}\sigma_1\sigma_2\). Correlation must lie between \(-1\) and \(1\), inclusive.

Isolating the Correlation Effect

When weights and volatilities are fixed, the first two terms of the variance formula do not change. Only the cross term changes with correlation. For positive weights, a higher correlation increases that term and therefore increases variance. Since standard deviation is the nonnegative square root of variance, it also increases as long as the variance changes upward.

The size of the variance change can be calculated directly. If correlation changes from \(\rho_a\) to \(\rho_b\), with weights and volatilities unchanged, subtract the two portfolio variance expressions. The fixed terms cancel, leaving:

\Delta \operatorname{Var}(R_p) =2w_1w_2\sigma_1\sigma_2(\rho_b-\rho_a).

For long positions in both assets, \(w_1w_2\) is positive. Thus an increase in correlation produces an increase in variance, and a decrease in correlation produces a decrease in variance. This conclusion concerns variance; the corresponding change in standard deviation is not generally a fixed percentage or a linear change because taking a square root is nonlinear.

Why the Formula Has a Correlation Term

The two-asset formula follows from the variance of a weighted sum. Expanding the definition of variance produces each asset's weighted variance and a term for how the two returns vary together. Substituting covariance as correlation times the two standard deviations gives the formula.

Write the portfolio return.
\(R_p=w_1R_1+w_2R_2\).

Expand its variance.
\(\operatorname{Var}(R_p)=w_1^2\operatorname{Var}(R_1)+w_2^2\operatorname{Var}(R_2)+2w_1w_2\operatorname{Cov}(R_1,R_2)\). The factor of two appears because the expansion contains both cross-products, and covariance is symmetric.

Express covariance using correlation.
By the definition of correlation, \(\operatorname{Cov}(R_1,R_2)=\rho_{12}\sigma_1\sigma_2\). Also, \(\operatorname{Var}(R_j)=\sigma_j^2\) for each asset \(j\).

Substitute and collect terms.
\(\operatorname{Var}(R_p)=w_1^2\sigma_1^2+w_2^2\sigma_2^2+2w_1w_2\rho_{12}\sigma_1\sigma_2\), as required.

This derivation also clarifies the model's limits. Correlation describes linear co-movement over the selected period; it does not by itself describe the full return distributions or guarantee how assets will move in a future period. The formula is an exact variance identity when the inputs refer to the same returns and period. If an input is an estimate, the resulting portfolio variance is an estimate too.

Boundary Correlations

The allowed correlation range has useful endpoints. Substituting \(\rho_{12}=1\) or \(\rho_{12}=-1\) into the variance formula gives a perfect-square expression. Taking its nonnegative square root yields the corresponding portfolio standard deviation.

\rho_{12}=1: \quad \operatorname{SD}(R_p)=|w_1\sigma_1+w_2\sigma_2|, \qquad \rho_{12}=-1: \quad \operatorname{SD}(R_p)=|w_1\sigma_1-w_2\sigma_2|.

The absolute values matter, particularly if a weight is negative. With positive weights, the plus-one boundary adds the two weighted standard deviations. At the minus-one boundary, the portfolio standard deviation is the absolute difference between them. These expressions are boundary checks for the general formula; they do not mean that every pair of assets can actually attain either boundary.

Worked Example: Measuring Risk Across Several Correlations

A hypothetical fund holds 60% in Asset A and 40% in Asset B. Their annual standard deviations are 20% and 10%, respectively. Compare portfolio standard deviation at correlations \(-1\), \(-0.5\), \(0\), \(0.5\), and \(1\). Assume the weights and volatilities remain fixed for every comparison.

Method. Use the two-asset variance formula in decimal units, then take the square root. The weights are \(w_1=0.60\), \(w_2=0.40\), and the volatilities are \(\sigma_1=0.20\), \(\sigma_2=0.10\). First calculate the fixed terms:

w_1^2\sigma_1^2=(0.60)^2(0.20)^2=0.36(0.04)=0.0144, \qquad w_2^2\sigma_2^2=(0.40)^2(0.10)^2=0.16(0.01)=0.0016.

The coefficient of correlation is:

2w_1w_2\sigma_1\sigma_2 =2(0.60)(0.40)(0.20)(0.10)=0.0096.

Therefore \(\operatorname{Var}(R_p)=0.0160+0.0096\rho_{12}\). Substituting each correlation gives:

At \(\rho_{12}=-1\), variance is \(0.0160-0.0096=0.0064\); standard deviation is \(\sqrt{0.0064}=0.0800\), or 8.00%.
At \(\rho_{12}=-0.5\), variance is \(0.0160-0.0048=0.0112\); standard deviation is \(\sqrt{0.0112}=0.1058\), or 10.58%.
At \(\rho_{12}=0\), variance is \(0.0160\); standard deviation is \(\sqrt{0.0160}=0.1265\), or 12.65%.
At \(\rho_{12}=0.5\), variance is \(0.0160+0.0048=0.0208\); standard deviation is \(\sqrt{0.0208}=0.1442\), or 14.42%.
At \(\rho_{12}=1\), variance is \(0.0160+0.0096=0.0256\); standard deviation is \(\sqrt{0.0256}=0.1600\), or 16.00%.

The results are rounded to four decimal places for standard deviation in decimal form, equivalent to two decimal places in percentages. At the boundaries, a check gives \(|(0.60)(0.20)-(0.40)(0.10)|=|0.12-0.04|=0.08\) for correlation \(-1\), and \(0.12+0.04=0.16\) for correlation \(1\). Both agree with the formula calculations. A common mistake is to change the weights while comparing correlations; that would change more than the correlation effect being measured.

Using the Formula for Comparisons and Targets

The same setup can compare two non-boundary correlations or solve for a correlation consistent with a stated risk. In either case, calculate the fixed terms and cross-term coefficient separately. This makes it easier to see which quantities change and to check whether a proposed correlation is within the permitted range.

Worked Example: Comparing Two Assumed Correlations

A hypothetical investment pool allocates 30% to Asset C and 70% to Asset D. Their annual standard deviations are 12% and 25%. Compare portfolio risk if their correlation is \(-0.4\) versus \(0.6\). Keep the weights and volatilities unchanged.

Method. Calculate each variance from the same fixed terms. In decimals, \(w_1=0.30\), \(w_2=0.70\), \(\sigma_1=0.12\), and \(\sigma_2=0.25\). The fixed terms are:

(0.30)^2(0.12)^2=0.09(0.0144)=0.001296, \qquad (0.70)^2(0.25)^2=0.49(0.0625)=0.030625.

The correlation coefficient is:

$$ 2(0.30)(0.70)(0.12)(0.25)=0.0126. $$

For \(\rho_{12}=-0.4\), the cross term is \(0.0126(-0.4)=-0.00504\). Thus portfolio variance is \(0.001296+0.030625-0.00504=0.026881\), and portfolio standard deviation is \(\sqrt{0.026881}=0.163954\), or 16.40% to two decimal places.

For \(\rho_{12}=0.6\), the cross term is \(0.0126(0.6)=0.00756\). Thus portfolio variance is \(0.001296+0.030625+0.00756=0.039481\), and portfolio standard deviation is \(\sqrt{0.039481}=0.198698\), or 19.87% to two decimal places.

The variance difference is \(0.039481-0.026881=0.012600\). The direct change formula gives \(0.0126[0.6-(-0.4)]=0.0126(1.0)=0.012600\), which checks the subtraction. The common mistake avoided here is treating a change in correlation as if it were the same change in standard deviation; it is variance that changes linearly with correlation in this fixed-input comparison.

Worked Example: Inferring Correlation From Portfolio Variance

A hypothetical portfolio holds equal weights in Assets E and F. Their annual standard deviations are 10% and 20%. An analysis reports portfolio variance of \(0.0181\) in decimal-return units. Find the implied correlation, then check whether the result is admissible.

Method. Substitute the known quantities into the variance formula and solve for correlation. Here \(w_1=w_2=0.50\), \(\sigma_1=0.10\), and \(\sigma_2=0.20\). The fixed terms are:

$$ (0.50)^2(0.10)^2+(0.50)^2(0.20)^2 =0.25(0.01)+0.25(0.04) =0.0025+0.0100=0.0125. $$

The coefficient of correlation is:

$$ 2(0.50)(0.50)(0.10)(0.20)=0.0100. $$

The equation to solve is \(0.0181=0.0125+0.0100\rho_{12}\). Subtracting \(0.0125\) gives \(0.0056=0.0100\rho_{12}\), so \(\rho_{12}=0.56\). This is admissible because \(-1\leq0.56\leq1\). As an additional check, substituting it back gives \(0.0125+0.0100(0.56)=0.0125+0.0056=0.0181\).

A common error is to put the reported variance where the correlation belongs or to divide by only one asset's standard deviation. Solving the full equation shows that the fixed terms must first be removed, and the remaining variance must then be divided by the entire cross-term coefficient.

Common Traps and Interpretation

Correlation is not covariance. Correlation is unitless and constrained to \([-1,1]\); covariance is measured in the product of the return units and need not lie in that interval. If the problem gives covariance, use it directly in the covariance version of the portfolio formula, or first convert it using \(\rho_{12}=\operatorname{Cov}(R_1,R_2)/(\sigma_1\sigma_2)\). Do not insert a covariance value as though it were a correlation.

A second trap is to assume that every increase in correlation must increase risk, regardless of the portfolio weights. The correlation coefficient in the formula is multiplied by \(2w_1w_2\sigma_1\sigma_2\). For positive volatilities, the direction of the effect depends on the sign of \(w_1w_2\). The usual increasing-risk conclusion applies to two positive holdings. If one weight is negative, the coefficient is negative, so an increase in correlation decreases the formula's variance, all else equal.

Finally, changing correlation changes portfolio variance without changing either asset's standalone variance. It does not change the weighted-average expected return when expected returns and weights are fixed. Keeping those effects separate is important when interpreting a portfolio comparison.

Key takeaway: With fixed weights and asset volatilities, the two-asset portfolio variance is linear in correlation. For two positive holdings, higher correlation means higher portfolio variance and standard deviation. At correlations \(1\) and \(-1\), the standard deviation is the absolute sum or absolute difference of the weighted asset standard deviations, respectively.

Check Your Understanding

Use decimal returns in the variance formula, show the fixed terms and correlation term, and round final standard deviations to two decimal places.

A portfolio has weights 25% and 75%, asset standard deviations 8% and 16%, and correlation \(0\). Find its variance and standard deviation. (Answer: variance \(0.0148\); standard deviation \(12.17\%\))
Two assets have equal weights, standard deviations 10% and 14%, and correlation \(0.5\). Find the portfolio standard deviation. (Answer: \(10.44\%\))
A portfolio has weights 70% and 30%, standard deviations 18% and 12%, and correlation \(-0.25\). Calculate its variance and standard deviation. (Answer: variance \(0.014904\); standard deviation \(12.21\%\))
Two equally weighted assets have standard deviations 10% and 20%. Compare portfolio standard deviation at correlations \(0.3\) and \(-0.3\), with no change to weights or volatilities. (Answer: \(12.45\%\) at correlation \(0.3\); \(9.75\%\) at correlation \(-0.3\))
A portfolio has weights 60% and 40%, standard deviations 20% and 15%, and positive correlation \(\rho\). What is the largest correlation for which portfolio standard deviation is no more than 15%? (Answer: \(\rho\leq0.3125\))

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